Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy theory or homology theory?
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In some sense, the (relatively new) field of directed algebraic topology represents an attempt to do this. This article by Marco Grandis includes a discussion (on page 8) of why bitopological spaces are too general to admit a good (directed) homotopy theory. 

